Abstract

We construct a new iterative scheme by hybrid methods and prove strong convergence theorem for approximation of a common fixed point of two countable families of closed relatively quasi-nonexpansive mappings which is also a solution to a system of equilibrium problems in a uniformly smooth
and strictly convex real Banach space with Kadec-Klee property using the properties of generalized -projection operator. Using this result, we discuss strong convergence theorem concerning variational inequality and convex minimization problems in Banach spaces. Our results extend many known recent results in the literature.

1. Introduction

Let be a real Banach space with dual and a nonempty, closed, and convex subset of . A mapping is called nonexpansive if
A point is called a fixed point of if . The set of fixed points of is denoted by .

We denote by the normalized duality mapping from to defined by
The following properties of are well known (the reader can consult [1–3] for more details). (1)If is uniformly smooth, then is norm-to-norm uniformly continuous on each bounded subset of . (2).(3)If is reflexive, then is a mapping from onto .(4)If is smooth, then is single valued.

Throughout this paper, we denote by the functional on defined by
It is obvious from (1.3) that

Definition 1.1. Let be a nonempty subset of , and let be a mapping from into . A point is said to be an asymptotic fixed point of if contains a sequence which converges weakly to and . The set of asymptotic fixed points of is denoted by . We say that a mapping is relatively nonexpansive (see, e.g., [4–9]) if the following conditions are satisfied:(R1),(R2),(R3).

If satisfies (R1) and (R2), then is said to be relatively quasi-nonexpansive. It is easy to see that the class of relatively quasi-nonexpansive mappings contains the class of relatively nonexpansive mappings. Many authors have studied the methods of approximating the fixed points of relatively quasi-nonexpansive mappings (see, e.g., [10–12] and the references cited therein). Clearly, in Hilbert space , relatively quasi-nonexpansive mappings and quasi-nonexpansive mappings are the same, for , and this implies that
The examples of relatively quasi-nonexpansive mappings are given in [11].

Let be a bifunction of into . The equilibrium problem (see, e.g., [13–25]) is to find such that
for all . We will denote the solutions set of (1.6) by . Numerous problems in physics, optimization, and economics reduce to find a solution of problem (1.6). The equilibrium problems include fixed point problems, optimization problems, and variational inequality problems as special cases (see, e.g., [26]).

In [7], Matsushita and Takahashi introduced a hybrid iterative scheme for approximation of fixed points of relatively nonexpansive mapping in a uniformly convex real Banach space which is also uniformly smooth: ,
They proved that converges strongly to , where .

In [27], Plubtieng and Ungchittrakool introduced the following hybrid projection algorithm for a pair of relatively nonexpansive mappings: ,
where , , , and are sequences in satisfying and and are relatively nonexpansive mappings and is the single-valued duality mapping on . They proved under the appropriate conditions on the parameters that the sequence generated by (1.8) converges strongly to a common fixed point of and .

In [9], Takahashi and Zembayashi introduced the following hybrid iterative scheme for approximation of fixed point of relatively nonexpansive mapping which is also a solution to an equilibrium problem in a uniformly convex real Banach space which is also uniformly smooth: , , ,
where is the duality mapping on . Then, they proved that converges strongly to , where .

Furthermore, in [28], Qin et al. introduced the following hybrid iterative algorithm for approximation of common fixed point of two countable families of closed relatively quasi-nonexpansive mappings in a uniformly convex and uniform smooth real Banach space:
They proved that the sequence converges strongly to a common fixed point of the countable families and of closed relatively quasi-nonexpansive mappings in a uniformly convex and uniformly smooth Banach space under some appropriate conditions on , , , and .

Recently, Li et al. [29] introduced the following hybrid iterative scheme for approximation of fixed points of a relatively nonexpansive mapping using the properties of generalized -projection operator in a uniformly smooth real Banach space which is also uniformly convex: , ,
They proved a strong convergence theorem for finding an element in the fixed points set of . We remark here that the results of Li et al. [29] extended and improved on the results of Matsushita and Takahashi [7].

Quite recently, motivated by the results of Takahashi and Zembayashi [9], Cholamjiak and Suantai [30] proved the following strong convergence theorem by hybrid iterative scheme for approximation of common fixed point of a countable family of closed relatively quasi-nonexpansive mappings which is also a solution to a system of equilibrium problems in uniformly convex and uniformly smooth Banach space.

Theorem 1.2. Let be a uniformly convex real Banach space which is also uniformly smooth, and let be a nonempty, closed, and convex subset of . For each , let be a bifunction from satisfying (A1)–(A4). Suppose is an infinitely countable family of closed and relatively quasi-nonexpansive mappings of into itself such that . Suppose is iteratively generated by ,
Assume that and are sequences which satisfy the following conditions:(i),(ii). Then, converges strongly to .

Motivated by the above-mentioned results and the on-going research, it is our purpose in this paper to prove a strong convergence theorem for two countable families of closed relatively quasi-nonexpansive mappings which is also a solution to a system of equilibrium problems in a uniformly smooth and strictly convex real Banach space with Kadec-Klee property using the properties of generalized -projection operator. Our results extend the results of Matsushita and Takahashi [7], Takahashi and Zembayashi [9], Qin et al. [28], Cholamjiak and Suantai [30], Li et al. [29], and many other recent known results in the literature.

2. Preliminaries

Let be a real Banach space. The modulus of smoothness of is the function defined by is uniformly smooth if and only if
Let dim . The modulus of convexity of is the function defined by is uniformly convex if, for any , there exists a such that if with , and , then . Equivalently, is uniformly convex if and only if for all . A normed space is called strictly convex if for all , , we have .

Let be a smooth, strictly convex, and reflexive real Banach space, and let be a nonempty, closed, and convex subset of . Following Alber [31], the generalized projection from onto is defined by
The existence and uniqueness of follows from the property of the functional and strict monotonicity of the mapping (see, e.g., [3, 31–34]). If is a Hilbert space, then is the metric projection of onto .

Next, we recall the concept of generalized -projector operator, together with its properties. Let be a functional defined as follows:
where , , is a positive number, and is proper, convex, and lower semicontinuous. From the definitions of and , it is easy to see the following properties:(i) is convex and continuous with respect to when is fixed,(ii) is convex and lower semicontinuous with respect to when is fixed.

Definition 2.1 (see Wu and Huang [35]). Let be a real Banach space with its dual . Let be a nonempty, closed, and convex subset of . We say that is a generalized -projection operator if

For the generalized -projection operator, Wu and Huang [35] proved the following theorem basic properties.

Lemma 2.2 (see Wu and Huang [35]). Let be a real reflexive Banach space with its dual . Let be a nonempty, closed, and convex subset of . Then, the following statements hold:(i) is a nonempty closed convex subset of for all ,(ii)if is smooth, then, for all if and only if
(iii)if is strictly convex and is positive homogeneous (i.e., for all such that where ), then is a single-valued mapping.

Fan et al. [36] showed that the condition is positive homogeneous which appeared in Lemma 2.2 can be removed.

Lemma 2.3 (see Fan et al. [36]). Let be a real reflexive Banach space with its dual and a nonempty, closed, and convex subset of . Then, if is strictly convex, then is a single-valued mapping.

Recall that is a single-valued mapping when is a smooth Banach space. There exists a unique element such that for each . This substitution in (2.5) gives
Now, we consider the second generalized -projection operator in a Banach space.

Definition 2.4. Let be a real Banach space and a nonempty, closed, and convex subset of . We say that is a generalized -projection operator if

Obviously, the definition of is a relatively quasi-nonexpansive mapping and is equivalent to,
.

Lemma 2.5 (see Li et al. [29]). Let be a Banach space, and let be a lower semicontinuous convex functional. Then, there exists and such that

We know that the following lemmas hold for operator .

Lemma 2.6 (see Li et al. [29]). Let be a nonempty, closed, and convex subset of a smooth and reflexive Banach space . Then, the following statements hold:(i) is a nonempty closed and convex subset of for all ,(ii)for all if and only if
(iii)if is strictly convex, then is a single-valued mapping.

The fixed points set of a relatively quasi-nonexpansive mapping is closed and convex as given in the following lemma.

Lemma 2.8 (see Chang et al. [37]). Let be a nonempty, closed, and convex subset of a uniformly smooth and strictly convex real Banach space which also has Kadec-Klee property. Let be a closed relatively quasi-nonexpansive mapping of into itself. Then, is closed and convex.

Also, this following lemma will be used in the sequel.

Lemma 2.9 (see Cho et al. [38]). Let be a uniformly convex real Banach space. For arbitrary , let and such that . Then, there exists a continuous strictly increasing convex function
such that, for every , the following inequality holds:

For solving the equilibrium problem for a bifunction , let us assume that satisfies the following conditions:(A1) for all ,(A2) is monotone, that is, for all ,(A3)for each ,(A4)for each is convex and lower semicontinuous.

Lemma 2.10 (see Blum and Oettli [26]). Let be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space , and let be a bifunction of into satisfying (A1)–(A4). Let and . Then, there exists such that

Lemma 2.11 (see Takahashi and Zembayashi [39]). Let be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space . Assume that satisfies (A1)–(A4). For and , define a mapping as follows:
for all . Then, the following hold: (1) is singlevalued,(2) is firmly nonexpansive-type mapping, that is, for any ,
(3),(4) is closed and convex.

Lemma 2.12 (see Takahashi and Zembayashi [39]). Let be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space . Assume that satisfies (A1)–(A4), and let . Then, for each and ,

For the rest of this paper, the sequence converges strongly to and will be denoted by as , converges weakly to and will be denoted by and we will assume that such that .

We recall that a Banach space has Kadec-Klee property if, for any sequence and with and , as . We note that every uniformly convex Banach space has the Kadec-Klee property. For more details on Kadec-Klee property, the reader is referred to [2, 33].

Lemma 2.13 (see Li et al. [29]). Let be a Banach space and . Let be a proper, convex, and lower semicontinuous mapping with convex domain . If is a sequence in such that and , then .

3. Main Results

Theorem 3.1. Let be a uniformly smooth and strictly convex real Banach space which also has Kadec-Klee property. Let be a nonempty, closed, and convex subset of . For each , let be a bifunction from satisfying (A1)–(A4). Suppose and are two countable families of closed relatively quasi-nonexpansive mappings of into itself such that . Let be a convex and lower semicontinuous mapping with , and suppose is iteratively generated by ,
with the conditions(i),(ii),(iii) for some ,(iv) satisfying . Then, converges strongly to .

Proof. We first show that is closed and convex. It is obvious that is closed and convex. Suppose is closed and convex for some . For each , we see that is equivalent to
By the construction of the set , we see that is closed and convex. Therefore, is also closed and convex. Hence, is closed and convex. By taking and for all , we obtain .We next show that . For , we have . Then, for each , we obtain
So, . This implies that . Therefore, is well defined.We now show that exists. Since is convex and lower semicontinuous, applying Lemma 2.5, we see that there exists and such that
It follows that
Since , it follows from (3.5) that
for each . This implies that is bounded and so is . By the construction of , we have that and . It then follows from Lemma 2.7 that
It is obvious that
and so is nondecreasing. It follows that the limit of exists.Now since is bounded in and is reflexive, we may assume that , and since is closed and convex for each , it is easy to see that for each . Again since , from the definition of , we obtain
Since
then we obtain
This implies that . By Lemma 2.13, we obtain . In view of Kadec-Klee property of , we have that .We next show that . We first show that . By the fact that and , we obtain
Now, (3.7) implies that
Taking the limit as in (3.13), we obtain
Therefore,
It then yields that . Since , we have
Hence,
This implies that is bounded in . Since is reflexive, and so is reflexive, we can then assume that . In view of reflexivity of , we see that . Hence, there exists such that . Since
taking the limit inferior of both sides of (3.18) and in view of weak lower semicontinuity of , we have
that is, . This implies that and so . It follows from and Kadec-Klee property of that . Note that is hemicontinuous; it yields that . It then follows from and Kadec-Klee property of that . Hence,
Since is uniformly norm-to-norm continuous on bounded sets and , we obtain
Since is bounded, so are , , and . Also, since is uniformly smooth, is uniformly convex. Then, from Lemma 2.9, we have
It then follows that
But
From and , we obtain
Using the condition , we have
By property of , we have . Since is also uniformly norm-to-norm continuous on bounded sets, we have
Similarly, we can show that
Since and are closed, we have .Next, we show that . Now, by Lemma 2.12, we obtain
It then yields that . Since , we have
Hence,
This implies that is bounded in . Since is reflexive, and so is reflexive, we can then assume that . In view of reflexivity of , we see that . Hence, there exists such that . Since
taking the limit inferior of both sides of (3.32) and in view of weak lower semicontinuity of , we have
that is, . This implies that and so . It follows from and Kadec-Klee property of that . Note that is hemicontinuous; it yields that . It then follows from and Kadec-Klee property of that . By the fact that is relatively nonexpansive and using Lemma 2.12 again, we have that
Observe that
Using (3.35) in (3.34), we obtain
It then yields that . Since , we have
This implies that is bounded in . Since is reflexive, we can then assume that . Since
taking the limit inferior of both sides of (3.38) and in view of weak lower semicontinuity of , we have
that is, . This implies that . It follows from and Kadec-Klee property of that
Similarly, . This further implies that
Also, since is uniformly norm-to-norm continuous on bounded sets and using (3.41), we obtain
Since ,
By Lemma 2.11, we have that for each
Furthermore, using (A2), we obtain
By (A4), (3.43), and , we have for each
For fixed , let for all . This implies that . This yields that . It follows from (A1) and (A4) that
and hence
From condition (A3), we obtain
This implies that . Thus, . Hence, we have .Finally, we show that . Since is a closed and convex set, from Lemma 2.6, we know that is single valued and denote . Since and , we have
We know that is convex and lower semicontinuous with respect to when is fixed. This implies that
From the definition of and , we see that . This completes the proof.

Take for all in Theorem 3.1, then and . Then we obtain the following corollary.

Corollary 3.2. Let be a uniformly smooth and strictly convex real Banach space which also has Kadec-Klee property. Let be a nonempty, closed, and convex subset of . For each , let be a bifunction from satisfying (A1)–(A4). Suppose and are two countable families of closed relatively quasi-nonexpansive mappings of into itself such that . Suppose is iteratively generated by ,